Taught By

Tim Roughgarden

Professor

Transcript

And let's begin with the idea of shortest paths. So, again I'll give you the movie graph. I'll give you Kevin Bacon as a starting point. What's the fewest number of hops, the fewest number of edges on a path that leads to, say, Jon Hamm? So some notation, I'm going to use DIST of V, to denote this shortest path distance. So with respect to a starting node S, the fewest number of hops or the fewest number of edges on a path that starts at S, and goes to V. And again you can define this in the same way for undirected graphs or directed graphs. In a directed graph, you always want to traverse arcs in the forward direction, in the correct direction. And to do this we just have to add a very small amount of extra code to the BFS code that I showed you earlier. It's just gonna be a very small constant overhead, and basically it just keeps track of what layer each node belongs to, and the layers are exactly tracking shortest path distances away from the starting point S. So what's the extra code. Well first in the initialization step, you set your preliminary estimate of the distance, the number of the shortest path distance from S to vertex V as well if V equals S, you know you can get from S to S on a path of length zero, the empty path. And if it's any other vertex all bets are off, you have no idea if there's a path to V at all. So let's just initially put plus infinity for all vertices other than the starting point. This is something we will of course revise once we actually discover a path to vertex V. And the only other extra code you have to add is, when you're considering, so when you take a vertex off of the front of the queue and then you iterate through its edges and you're considering one of those edges V, W, so your V would be the vertex that you just removed from the front of the queue. And as usual if the other end of the edge W has already been dealt with then, you know, you just throw it out. That would be redundant work to look at it again. But if this is the first time you've seen the vertex W. Then, in addition to what we did previously, in addition to marking it as explored and putting it in the queue at the back, we also compute its distance, and its distance is just going to be one more than the distance of the vertex V, responsible for W's addition to the queue, responsible for first discovering this vertex W. So, returning to our running example of breadth first search, let's see what happens. So, again, remember the way this worked is we start out with from the vertex S, and we set the distance, you know in our initialization equal to zero. We don't know what the distance is of anything else. So, then how did breadth first search work? So, we, in the initial step we put S in the queue. We go to the main while loop, and then the queue's not empty. We extract S from the queue. We look at its neighbors. Those neighbors are A and B. We handle them in some order. Let's again think of that we first handle the edge between S and A. So, then what do we do? We say we haven't seen A yet. So we mark A as explored. We put A in the queue at the front, and now we have this extra step. It's the first time we're seeing A, so we wanna compute its distance. And we compute its distance as one more than the vertex responsible for discovering A. And so in this case S was the vertex whose exploration unveiled the existence of the vertex A to us. S's distance is zero so we set A's distance to one. And that's tantamount to being a member of the ith layer. So what happens in the next iteration of the while loop. So now the queue contains Sorry, the next iteration of the for loop, excuse me. So after we've handled the edge S comma A, we're still dealing with S's edges, now we handle the edge S comma B. We put, it is the first time we've seen B. We put B at the end of the queue, we mark it as explored, and then we also execute this new step. We set B's distance to one more than the vertex responsible for discovering it. That would again be the vertex S. S led to B's discovery. And so we set B's distance to be one more than S's distance, also known as one. And that corresponds to being the other node in layer one. Now having handled all of S's adjacent arcs we go back to the while loop. We ask if the queue is empty. Certainly not. It takes two vertices, first A then B. We extract the first vertex cuz it's FIFO, that would be the vertex A. Now we look at A's incident edges. There's S comma A, which we ignore. There's A comma C. This is the first time we've seen C. So as before we mark C as explored. We add C to the end of the queue and now again we have this additional line. We set C's distance to be one more than the vertex responsible for its discovery. In this case it's A. That first discovered C. So we're gonna set C's distance to be one more than A's distance also known as two. So then having handled A we move on to the next vertex in the queue, which in this case is B. Again we can forget about the edge between S and V. We've already seen S, we can forget about the edge between B and C. We've already seen C but D is now discovered for the first time via B. It gets more as explored, it goes to the end of the queue and its distance is set equal to one more than B's distances which is two. So, then we deal with C. Again it has four arcs, four edges, three of them are irrelevant. The one to E is not irrelevant, cause this is the first time we've seen E. So, E's distance is computed as one more than C, cause C was the one who first found E, and so E gets a distance of three, and then the rest of the algorithm proceeds as before. And you will notice that the labelings, the shortest path labels, are exactly the layers as promised. I hope you find it very easy to believe at this point that, that claim is true in general. That the distance computed by breadth-first search for an arbitrary vertex V, that's reachable from S is, that's gonna be equal to i if and only if V is in the ith layer as we've been defining it previously. And what does it really mean to be in the ith layer? It means that the shortest path distance between V and S has i hops, i edges. So I don't wanna spend time giving a super rigorous proof of this claim but let me just give you the gist, the basic idea, and I encourage you to produce some formal proof at home if that is something that interests you. So one way to do it is you can do it by induction on the layer i. And so what you want to prove is that all of the nodes that belong to a given layer i do Indeed, breadth first search does indeed compute the distance of i for them. So what does it mean to be a node in layer i? Well, first of all, you can't have been seen in either of the, any of the previous layers; you weren't a member of layer zero through i minus one. And furthermore, you're a, a neighbor of somebody who's in layer i minus one. Right? You're seen for the first time once all of the layer i minus one nodes are processed. So the inductive hypothesis tells you that distances were correctly computed for everybody from the lower l, from the lower layers. So in particular, whoever this node V was from layer i minus one was responsible for discovering u, in layer i. It has a distance computed as i minus one. Yours is assigned to be one more than its, namely i. So that pushes through the inductive step everything in layer i indeed gets the correct label of a shortest path distance i away from S. So before we wrap up with this application, I do wanna emphasize, it is only breadth first search that gives us this guarantee of shortest paths. So, we have a wide family of graph search strategies, all of which find everything findable. Breadth-first search is one of those, but this is a special additional property that breadth-first search has: you get shortest path distances from it. So in particular depth-first search does not in general compute shortest path distances. This is really a special property of breadth-first search. By contrast in this next application, which is going to be computing the connected components of an undirected graph, this is not really fundamental to breadth first search. For example, you could use depth first search instead and that would work just as well.

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